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Risks, Volume 5, Issue 1 (March 2017)

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Editorial

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Open AccessEditorial Acknowledgement to Reviewers of Risks in 2016
Risks 2017, 5(1), 4; doi:10.3390/risks5010004
Received: 12 January 2017 / Revised: 12 January 2017 / Accepted: 12 January 2017 / Published: 12 January 2017
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Abstract
The editors of Risks would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016. [...]
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Research

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Open AccessArticle Optimal Retention Level for Infinite Time Horizons under MADM
Risks 2017, 5(1), 1; doi:10.3390/risks5010001
Received: 26 September 2016 / Revised: 16 December 2016 / Accepted: 19 December 2016 / Published: 27 December 2016
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Abstract
In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum
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In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum benefit criteria: released capital, expected profit and exponential-fractional-logarithmic utility from the insurer’s point of view. Numerical examples are presented to explain how the optimal initial surplus and retention level are changed according to the individual claim amounts, loading factors and weights of the criteria. In the decision making process, we use The Analytical Hierarchy Process (AHP) and The Technique for Order of Preference by Similarity to ideal Solution (TOPSIS) methods as the Multi-Attribute Decision Making methods (MADM) and compare our results considering different combinations of loading factors for both exponential and Pareto individual claims. Full article
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Open AccessFeature PaperArticle On Comparison of Stochastic Reserving Methods with Bootstrapping
Risks 2017, 5(1), 2; doi:10.3390/risks5010002
Received: 30 September 2016 / Revised: 19 December 2016 / Accepted: 20 December 2016 / Published: 4 January 2017
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Abstract
We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping
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We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping framework, we discuss the choice of residuals, namely the Pearson residuals, the deviance residuals and the Anscombe residuals. In addition, several possible residual adjustments are discussed and compared in a case study. We carry out a practical implementation and comparison of methods using real-life insurance data to estimate reserves and their prediction errors. We propose to consider proper scoring rules for model validation, and the assessments will be drawn from an extensive case study. Full article
Open AccessFeature PaperArticle The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation
Risks 2017, 5(1), 3; doi:10.3390/risks5010003
Received: 21 November 2016 / Revised: 22 December 2016 / Accepted: 28 December 2016 / Published: 6 January 2017
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Abstract
We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary
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We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this. Full article
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Open AccessFeature PaperArticle Minimum Protection in DC Funding Pension Plans and Margrabe Options
Risks 2017, 5(1), 5; doi:10.3390/risks5010005
Received: 14 November 2016 / Revised: 22 December 2016 / Accepted: 10 January 2017 / Published: 18 January 2017
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Abstract
The regulation on the Belgian occupational pension schemes has been recently changed. The new law allows for employers to choose between two different types of guarantees to offer to their affiliates. In this paper, we address the question arising naturally: which of the
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The regulation on the Belgian occupational pension schemes has been recently changed. The new law allows for employers to choose between two different types of guarantees to offer to their affiliates. In this paper, we address the question arising naturally: which of the two guarantees is the best one? In order to answer that question, we set up a stochastic model and use financial pricing tools to compare the methods. More specifically, we link the pension liabilities to a portfolio of financial assets and compute the price of exchange options through the Margrabe formula. Full article
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Open AccessFeature PaperArticle Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model
Risks 2017, 5(1), 6; doi:10.3390/risks5010006
Received: 9 October 2016 / Revised: 20 December 2016 / Accepted: 12 January 2017 / Published: 22 January 2017
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Abstract
We consider an insurer who faces an external jump-diffusion risk that is negatively correlated with the capital returns in a multidimensional regime switching model. The insurer selects investment and liability ratio policies continuously to maximize her/his expected utility of terminal wealth. We obtain
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We consider an insurer who faces an external jump-diffusion risk that is negatively correlated with the capital returns in a multidimensional regime switching model. The insurer selects investment and liability ratio policies continuously to maximize her/his expected utility of terminal wealth. We obtain explicit solutions of optimal policies for logarithmic and power utility functions. We study the impact of the insurer’s risk aversion, the negative correlation between the external risk and the capital returns, and the regime of the economy on the optimal policy. We find, among other things, that the regime of the economy and the negative correlation between the external risk and the capital returns have a dramatic effect on the optimal policy. Full article
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Open AccessArticle Change Point Estimation in Panel Data without Boundary Issue
Risks 2017, 5(1), 7; doi:10.3390/risks5010007
Received: 28 August 2016 / Revised: 22 December 2016 / Accepted: 17 January 2017 / Published: 22 January 2017
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Abstract
Panel data of our interest consist of a moderate number of panels, while the panels contain a small number of observations. An estimator of common breaks in panel means without a boundary issue for this kind of scenario is proposed. In particular, the
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Panel data of our interest consist of a moderate number of panels, while the panels contain a small number of observations. An estimator of common breaks in panel means without a boundary issue for this kind of scenario is proposed. In particular, the novel estimator is able to detect a common break point even when the change happens immediately after the first time point or just before the last observation period. Another advantage of the elaborated change point estimator is that it results in the last observation in situations with no structural breaks. The consistency of the change point estimator in panel data is established. The results are illustrated through a simulation study. As a by-product of the developed estimation technique, a theoretical utilization for correlation structure estimation, hypothesis testing and bootstrapping in panel data is demonstrated. A practical application to non-life insurance is presented, as well. Full article
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Open AccessArticle n-Dimensional Laplace Transforms of Occupation Times for Spectrally Negative Lévy Processes
Risks 2017, 5(1), 8; doi:10.3390/risks5010008
Received: 30 November 2016 / Revised: 3 January 2017 / Accepted: 17 January 2017 / Published: 29 January 2017
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Abstract
Using a Poisson approach, we find Laplace transforms of joint occupation times over n disjoint intervals for spectrally negative Lévy processes. They generalize previous results for dimension two. Full article
Open AccessArticle The Shifting Shape of Risk: Endogenous Market Failure for Insurance
Risks 2017, 5(1), 9; doi:10.3390/risks5010009
Received: 6 July 2016 / Accepted: 9 December 2016 / Published: 27 January 2017
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Abstract
This article considers an economy where risk is insurable, but selection determines the pool of individuals who take it up. First, we demonstrate that the comparative statics of these economies do not necessarily depend on its marginal selection (adverse versus favorable), but rather
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This article considers an economy where risk is insurable, but selection determines the pool of individuals who take it up. First, we demonstrate that the comparative statics of these economies do not necessarily depend on its marginal selection (adverse versus favorable), but rather other characteristics. We then use repeated cross-sections of medical expenditures in the U.S. to understand the role of changes in the medical risk distribution on the fraction of Americans without medical insurance. We find that both the level and the shape of the distribution of risk are important in determining the equilibrium quantity of insurance. Symmetric changes in risk (e.g., shifts in the price of medical care) better explain the shifting insurance rate over time. Asymmetric changes (e.g., associated with a shifting age distribution) are not as important. Full article
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Open AccessFeature PaperArticle Distinguishing Log-Concavity from Heavy Tails
Risks 2017, 5(1), 10; doi:10.3390/risks5010010
Received: 14 November 2016 / Revised: 10 January 2017 / Accepted: 17 January 2017 / Published: 7 February 2017
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Abstract
Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X1+X2 occur as result of a
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Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X 1 + X 2 occur as result of a single big jump with heavy tails whereas X 1 , X 2 are of equal order of magnitude in the light-tailed case. The method is based on the ratio | X 1 X 2 | / ( X 1 + X 2 ) , for which sharp asymptotic results are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot. Full article
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Open AccessFeature PaperArticle Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account
Risks 2017, 5(1), 11; doi:10.3390/risks5010011
Received: 18 November 2016 / Revised: 4 January 2017 / Accepted: 17 January 2017 / Published: 5 February 2017
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Abstract
Optimal forms of reinsurance policies have been studied for a long time in the actuarial literature. Most existing results are from the insurer’s point of view, aiming at maximizing the expected utility or minimizing the risk of the insurer. However, as pointed out
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Optimal forms of reinsurance policies have been studied for a long time in the actuarial literature. Most existing results are from the insurer’s point of view, aiming at maximizing the expected utility or minimizing the risk of the insurer. However, as pointed out by Borch (1969), it is understandable that a reinsurance arrangement that might be very attractive to one party (e.g., the insurer) can be quite unacceptable to the other party (e.g., the reinsurer). In this paper, we follow this point of view and study forms of Pareto-optimal reinsurance policies whereby one party’s risk, measured by its value-at-risk (VaR), cannot be reduced without increasing the VaR of the counter-party in the reinsurance transaction. We show that the Pareto-optimal policies can be determined by minimizing linear combinations of the VaR s of the two parties in the reinsurance transaction. Consequently, we succeed in deriving user-friendly, closed-form, optimal reinsurance policies and their parameter values. Full article
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Open AccessArticle A Discussion of a Risk-Sharing Pension Plan
Risks 2017, 5(1), 12; doi:10.3390/risks5010012
Received: 2 October 2016 / Revised: 22 November 2016 / Accepted: 27 January 2017 / Published: 14 February 2017
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Abstract
I show that risk-sharing pension plans can reduce some of the shortcomings of defined benefit and defined contributions plans. The risk-sharing pension plan presented aims to improve the stability of benefits paid to generations of members, while allowing them to enjoy the expected
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I show that risk-sharing pension plans can reduce some of the shortcomings of defined benefit and defined contributions plans. The risk-sharing pension plan presented aims to improve the stability of benefits paid to generations of members, while allowing them to enjoy the expected advantages of a risky investment strategy. The plan does this by adjusting the investment strategy and benefits in response to a changing funding level, motivated by the with-profits contract proposed by Goecke (2013). He suggests a mean-reverting log reserve (or funding) ratio, where mean reversion occurs through adjustments to the investment strategy and declared bonuses. To measure the robustness of the plan to human factors, I introduce a measurement of disappointment, where disappointment is high when there are many consecutive years over which benefit payments are declining. Another measure introduced is devastation, where devastation occurs when benefit payments are zero. The motivation is that members of a pension plan who are easily disappointed or likely to get no benefit, are more likely to exit the plan. I find that the risk-sharing plan offers more disappointment than a defined contribution plan, but it eliminates the devastation possible in a plan that tries to accumulate contributions at a steadily increasing rate. The proposed risk-sharing plan can give a narrower range of benefits than in a defined contribution plan. Thus it can offer a stable benefit to members without the risk of running out of money. Full article
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Open AccessArticle Mathematical Analysis of Replication by Cash Flow Matching
Risks 2017, 5(1), 13; doi:10.3390/risks5010013
Received: 17 August 2016 / Accepted: 24 February 2017 / Published: 28 February 2017
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Abstract
The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note,weelaborateononespecificformulationofareplicatingportfolioproblem. Incontrasttothetwo most popular replication approaches, it does not yield an analytic solution (if, at
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The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note,weelaborateononespecificformulationofareplicatingportfolioproblem. Incontrasttothetwo most popular replication approaches, it does not yield an analytic solution (if, at all, a solution exists andisunique). Further,althoughconvex,theobjectivefunctionseemstobenon-smooth,andhencea numericalsolutionmightthusbemuchmoredemandingthanforthetwomostpopularformulations. Especially for the second reason, this formulation did not (yet) receive much attention in practical applications, in contrast to the other two formulations. In the following, we will demonstrate that the (potential) non-smoothness can be avoided due to an equivalent reformulation as a linear second order cone program (SOCP). This allows for a numerical solution by efficient second order methods like interior point methods or similar. We also show that—under weak assumptions—existence and uniqueness of the optimal solution can be guaranteed. We additionally prove that—under a further similarly weak condition—the fair value of the replicating portfolio equals the fair value of liabilities. Based on these insights, we argue that this unloved stepmother child within the replication problem family indeed represents an equally good formulation for practical purposes. Full article
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Open AccessArticle Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks
Risks 2017, 5(1), 14; doi:10.3390/risks5010014
Received: 28 November 2016 / Accepted: 24 February 2017 / Published: 3 March 2017
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Abstract
This paper considered a dependent discrete-time risk model, in which the insurance risks are represented by a sequence of independent and identically distributed real-valued random variables with a common Gamma-like tailed distribution; the financial risks are denoted by another sequence of independent and
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This paper considered a dependent discrete-time risk model, in which the insurance risks are represented by a sequence of independent and identically distributed real-valued random variables with a common Gamma-like tailed distribution; the financial risks are denoted by another sequence of independent and identically distributed positive random variables with a finite upper endpoint, but a general dependence structure exists between each pair of the insurance risks and the financial risks. Following the works of Yang and Yuen in 2016, we derive some asymptotic relations for the finite-time and infinite-time ruin probabilities. As a complement, we demonstrate our obtained result through a Crude Monte Carlo (CMC) simulation with asymptotics. Full article
Open AccessFeature PaperArticle Change Point Detection and Estimation of the Two-Sided Jumps of Asset Returns Using a Modified Kalman Filter
Risks 2017, 5(1), 15; doi:10.3390/risks5010015
Received: 31 August 2016 / Revised: 15 January 2017 / Accepted: 27 February 2017 / Published: 3 March 2017
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Abstract
In the first part of the paper, the positive and negative jumps of NASDAQ daily (log-) returns and three of its stocks are estimated based on the methodology presented by Theodosiadou et al. 2016, where jumps are assumed to be hidden random variables.
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In the first part of the paper, the positive and negative jumps of NASDAQ daily (log-) returns and three of its stocks are estimated based on the methodology presented by Theodosiadou et al. 2016, where jumps are assumed to be hidden random variables. For that reason, the use of stochastic state space models in discrete time is adopted. The daily return is expressed as the difference between the two-sided jumps under noise inclusion, and the recursive Kalman filter algorithm is used in order to estimate them. Since the estimated jumps have to be non-negative, the associated pdf truncation method, according to the non-negativity constraints, is applied. In order to overcome the resulting underestimation of the empirical time series, a scaling procedure follows the stage of truncation. In the second part of the paper, a nonparametric change point analysis concerning the (variance–) covariance is applied to the NASDAQ return time series, as well as to the estimated bivariate jump time series derived after the scaling procedure and to each jump component separately. A similar change point analysis is applied to the three other stocks of the NASDAQ index. Full article
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Open AccessArticle Evaluating Extensions to Coherent Mortality Forecasting Models
Risks 2017, 5(1), 16; doi:10.3390/risks5010016
Received: 25 October 2016 / Accepted: 10 February 2017 / Published: 10 March 2017
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Abstract
Coherent models were developed recently to forecast the mortality of two or more sub-populations simultaneously and to ensure long-term non-divergent mortality forecasts of sub-populations. This paper evaluates the forecast accuracy of two recently-published coherent mortality models, the Poisson common factor and the product-ratio
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Coherent models were developed recently to forecast the mortality of two or more sub-populations simultaneously and to ensure long-term non-divergent mortality forecasts of sub-populations. This paper evaluates the forecast accuracy of two recently-published coherent mortality models, the Poisson common factor and the product-ratio functional models. These models are compared to each other and the corresponding independent models, as well as the original Lee–Carter model. All models are applied to age-gender-specific mortality data for Australia and Malaysia and age-gender-ethnicity-specific data for Malaysia. The out-of-sample forecast error of log death rates, male-to-female death rate ratios and life expectancy at birth from each model are compared and examined across groups. The results show that, in terms of overall accuracy, the forecasts of both coherent models are consistently more accurate than those of the independent models for Australia and for Malaysia, but the relative performance differs by forecast horizon. Although the product-ratio functional model outperforms the Poisson common factor model for Australia, the Poisson common factor is more accurate for Malaysia. For the ethnic groups application, ethnic-coherence gives better results than gender-coherence. The results provide evidence that coherent models are preferable to independent models for forecasting sub-populations’ mortality. Full article
(This article belongs to the Special Issue Ageing Population Risks)
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Open AccessArticle The Impact of Changes to the Unemployment Rate on Australian Disability Income Insurance Claim Incidence
Risks 2017, 5(1), 17; doi:10.3390/risks5010017
Received: 7 February 2017 / Revised: 6 March 2017 / Accepted: 8 March 2017 / Published: 14 March 2017
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Abstract
We explore the extent to which claim incidence in Disability Income Insurance (DII) is affected by changes in the unemployment rate in Australia. Using data from 1986 to 2001, we fit a hurdle model to explore the presence and magnitude of the effect
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We explore the extent to which claim incidence in Disability Income Insurance (DII) is affected by changes in the unemployment rate in Australia. Using data from 1986 to 2001, we fit a hurdle model to explore the presence and magnitude of the effect of changes in unemployment rate on the incidence of DII claims, controlling for policy holder characteristics and seasonality. We find a clear positive association between unemployment and claim incidence, and we explore this further by gender, age, deferment period, and occupation. A multinomial logistic regression model is fitted to cause of claim data in order to explore the relationship further, and it is shown that the proportion of claims due to accident increases markedly with rising unemployment. The results suggest that during periods of rising unemployment, insurers may face increased claims from policy holders with shorter deferment periods for white-collar workers and for medium and heavy manual workers. Our findings indicate that moral hazard may have a material impact on DII claim incidence and insurer business in periods of declining economic conditions. Full article
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Open AccessFeature PaperArticle Context Moderates Priming Effects on Financial Risk Taking
Risks 2017, 5(1), 18; doi:10.3390/risks5010018
Received: 3 December 2016 / Revised: 6 March 2017 / Accepted: 9 March 2017 / Published: 14 March 2017
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Abstract
Previous research has shown that risk preferences are sensitive to the financial domain in which they are framed. In the present paper, we explore whether the effect of negative priming on risk taking is moderated by financial context. A total of 120 participants
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Previous research has shown that risk preferences are sensitive to the financial domain in which they are framed. In the present paper, we explore whether the effect of negative priming on risk taking is moderated by financial context. A total of 120 participants completed questionnaires, where risky choices were framed in six different financial scenarios. Half of the participants were allocated to a negative priming condition. Negative priming reduced risk-seeking behaviour compared to a neutral condition. However, this effect was confined to non-experiential scenarios (i.e., gamble to win, possibility to lose), and not to ‘real world’ financial products (e.g., pension provision). The results call into question the generalisability of priming effects on different financial contexts. Full article
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Open AccessFeature PaperArticle Immunization and Hedging of Post Retirement Income Annuity Products
Risks 2017, 5(1), 19; doi:10.3390/risks5010019
Received: 17 January 2017 / Revised: 5 March 2017 / Accepted: 13 March 2017 / Published: 16 March 2017
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Abstract
Designing post retirement benefits requires access to appropriate investment instruments to manage the interest rate and longevity risks. Post retirement benefits are increasingly taken as a form of income benefit, either as a pension or an annuity. Pension funds and life insurers offer
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Designing post retirement benefits requires access to appropriate investment instruments to manage the interest rate and longevity risks. Post retirement benefits are increasingly taken as a form of income benefit, either as a pension or an annuity. Pension funds and life insurers offer annuities generating long term liabilities linked to longevity. Risk management of life annuity portfolios for interest rate risks is well developed but the incorporation of longevity risk has received limited attention. We develop an immunization approach and a delta-gamma based hedging approach to manage the risks of adverse portfolio surplus using stochastic models for mortality and interest rates. We compare and assess the immunization and hedge effectiveness of fixed-income coupon bonds, annuity bonds, as well as longevity bonds, using simulations of the portfolio surplus for an annuity portfolio and a range of risk measures including value-at-risk. We show how fixed-income annuity bonds can more effectively match cash flows and provide additional hedge effectiveness over coupon bonds. Longevity bonds, including deferred longevity bonds, reduce risk significantly compared to coupon and annuity bonds, reflecting the long duration of the typical life annuity and the exposure to longevity risk. Longevity bonds are shown to be effective in immunizing surplus over short and long horizons. Delta gamma hedging is generally only effective over short horizons. The results of the paper have implications for how providers of post retirement income benefit streams can manage risks in demanding conditions where innovation in investment markets can support new products and increase the product range. Full article
(This article belongs to the Special Issue Designing Post-Retirement Benefits in a Demanding Scenario)
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Open AccessArticle Optimal Time to Enter a Retirement Village
Risks 2017, 5(1), 20; doi:10.3390/risks5010020
Received: 14 October 2016 / Revised: 23 January 2017 / Accepted: 18 March 2017 / Published: 22 March 2017
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Abstract
We consider the financial planning problem of a retiree wishing to enter a retirement village at a future uncertain date. The date of entry is determined by the retiree’s utility and bequest maximisation problem within the context of uncertain future health states. In
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We consider the financial planning problem of a retiree wishing to enter a retirement village at a future uncertain date. The date of entry is determined by the retiree’s utility and bequest maximisation problem within the context of uncertain future health states. In addition, the retiree must choose optimal consumption, investment, bequest and purchase of insurance products prior to their full annuitisation on entry to the retirement village. A hyperbolic absolute risk-aversion (HARA) utility function is used to allow necessary consumption for basic living and medical costs. The retirement village will typically require an initial deposit upon entry. This threshold wealth requirement leads to exercising the replication of an American put option at the uncertain stopping time. From our numerical results, active insurance and annuity markets are shown to be a critical aspect in retirement planning. Full article
(This article belongs to the Special Issue Ageing Population Risks)
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